The structure of bonds in biomolecules, such as base pairs in RNA chains or native interactions in proteins, can be presented in the form of a chord diagram. A given biomolecule is then characterized by the genus of an auxiliary two-dimensional surface associated to such a diagram. In this work we introduce the notion of the genus trace, which describes dependence of genus on the choice of a subchain of a given backbone chain. We find that the genus trace encodes interesting physical and biological information about a given biomolecule and its three dimensional structural complexity; in particular it gives a way to quantify how much more complicated a biomolecule is than its nested secondary structure alone would indicate. We illustrate this statement in many examples, involving both RNA and protein chains. First, we conduct a survey of all published RNA structures with better than 3 Å resolution in the PDB database, and find that the genus of natural structural RNAs has roughly linear dependence on their length. Then, we show that the genus trace captures properties of various types of base pairs in RNA, and enables the identification of the domain structure of a ribosome. Furthermore, we find that not only does the genus trace detect a domain structure, but it also predicts a cooperative folding pattern in multi-domain proteins. The genus trace turns out to be a useful and versatile tool, with many potential applications.